Revista Mexicana de Física 42, Suplemento 1 (1996) 152-162

Partial symmetries in nuclear spectroscopy

A. LEVIATAN

Racah Institute of Physics, The Hebrew UniversityJerusa/em 91904, Israel

ABSTRACT. The notions of exact, dynamical and partial syrnmetries are discussed in relation tonudear spectroscopy. Explicil forms of Hamiltonians with partial SU(3) symmetry are presenledin the framework of lhe interacting boson model of nudei. An analysis of the resulting spectrumand electromagnetic transilions demonslrales lhe relevance of sueh parlial symmetry lo lhe spec-lroscopy ofaxially deformed nudeL

RESUMEN. Se discuten las nociones de simetría exacta y dinámica en relación con espectroscopÍanuclear. Se presentan formas explícitas de los hamiltonianos con simetría espacial SU(3), en el con-texto del modelo nuclear de bosones interactuantes. Un análisis de los espectros resultantes y de lastransiciones electromagnéticas demuestra la relevancia de tal simetría parcial en la espectroscopÍade núcleos con deformación axial.

PACS: 21.60.Fw; 21.10.Re; 21.60.Ev

1. INTRODUCTION

Symmetry plays an important role in analyzing spectroscopic properties of nuelei. It pro-vides quantum numbers to elassify the eigenstates of the system and allows the use ofthe \Vigner-Eckart theorem to obtain selection rules and to facilitate evaluation of ma-trix elements. When an exact symmetry occurs, the Hamiltonian commutes with all thegenerators of the symmetry group and consequently admits a block structure in a basislabeled by the corresponding irreducible representations (irreps). Eigenstates which be-long to the same irrep of the symmetry group are degenerate. In a dynamical symmetrythe Hamiltonian is written in terms of the Casimir operators of a chain of subgroups. Thelabels of irreps of the groups in the chain serve as quantum numbers to elassify all theeigenstates of the Hamiltonian. Sta tes belonging to a given irrep are no longer degeneratebut different irreps are not mixed and the Hamiltonian retains its block structure. Thecorresponding wave-fnnctions are determined solely by the symmetry and are thereforeindependent of parameters in the Hamiltonian. A great advantage of having a dynamicalsymmetry is that it enables a derivation of elosed analytic expressions for the eigenstates,eigenvalues and other observables of the system (e.y., transition rates).The merits of a (dynamical) symmetry are self-evident. However, in detailed applica-

tions of group theoretical schemes to the spectroscopy of nuelei, one often finds that theassumed symmetry is fulfilled by only sorne of the states but not by other. For exam-pie, certain degeneracies implied by the symmetry are not observed experimentally, asis the case with 8U(3) assignments of rotational bands in deformed nuelei. These andother observations motivate one to consider a particular breaking of the symmetry thatwould result in mixing of representations in some part of the spectrum while retaining a

152

PARTIALSYMMETRIESIN NUCLEARSPECTROSCOPY 153

good symmetry to specific eigenstates. We refer to such a situation as partial (dynamical)symmetry. Within such symmetry construction only a subset of eigenstates are pure andpreserve the desired features of a dynamical symmetry (e.g., solvability).Dynamical symmetries received considerable attention in nuelear physics with the in-

troduction of the interacting boson model (IBM). The empirical success, computationalsimplicity and inherent symmetry structure of the model provide a convenient environ-ment to study the different notions of symmetries, in relation to nuelear spectroscopy. Inthis context we review briefly the symmetries of the 113M, and then focus the discussionto the 5U(3) symmetry which is relevant to axially deformed nuelei. Motivating the needfor the symmetry to be partial, we proceed to construct 113M Hamiltonians with suchproperty, and apply the scheme to the spectroscopy of 168Er.

2. EXACT SYMMETRIES IN TllE 113M

The interacting boson model (IBM) [11 describes collective states in even-even nuelei interms of an assembly of a fixed number N of bosons. The bosons are of two kinds, s and d,with angular momentum zero and two respectively. In its simplest form the Hamiltonianconsists of hermitian, scalar, one- and two-body interactions which conserve the totalnumber of bosons IV = sI s + L~dld~. Accordingly, the Hamiltonian is expanded in termsof number-conserving bilinear products of boson operators. The latter are the generatorsof U(6) which serves as a spectrum generating algebra. The IBM exhibits three dynamicalsymmetries connected with chains of subgroups of the U(6) group that inelude the 0(3)rotation subgroup

U(6) ::J U(5) ::J 0(5) ::J 0(3) anharmonic vibrator,

U(6) ::J 5U(3) ::J 0(3) axial rotor,

U(6) ::J 0(6) ::J 0(5) ::J 0(3) -y-unstable rotor.

(1)

5uch limiting cases provide analytic solutions whose from resembles known geometricmodels, as indicated in Eq. (1). The general Hamiltonian is a mixture of the Casimiroperators of all of the aboye groups. It can be diagonalized in any one of the completebases attached to each chain. A dynamical syrnmetry correspond to the special case whenthe Hamiltonian contains only the Casimir operators of a single chain.The natural starting point for the mM description ofaxially deformed nuelei is the

5U(3) dynamical symmetry, corresponding to the chain

U(6) ::J 5U(3) ::J 0(3)! 1 1

INI (.\,¡L) J( LM.(2)

The basis states are labeled by I[N](.\, ¡L)J( Llvf), where N is the total nurnber of bosons,L the angular momentul11, (.\,1') denote the 5U(3) irreps and J( is an additional label

154 A. LEVIATAN

needed for complete elassification. The Hamiltonian in this case may be transcribed in theform

H(5U(3)) = h2 [-CSU(3) + 2Ñ(2lV + 3)] + ACO(3)' (3)

The ,V and Ñ2 terms are constants fixing the ground state at zero energy. Ce denotesthe Casimir operator of the group G, CSU(3) = 2Q(2) . Q(2) + (3/4)L(1) . L(I), and CO(3) =L(1).L(1) with eigenvalues (A2+1'2+A11+3A+31') and L(L+l) respectively. The generatorsof 5U(3) are the quadrupole Q(2) and angular momentum L(1) operators

(4)

The al!owed 5U(3) representations are (A,I') = (2N - 4k - 6m, 2k) with A, 1', m, k non-negative integers. The corresponding spectrum of H(5U(3)) is

E = 6h2 [(2N + 1- 2k)k +(4N +3 - 6k - 6m)ml + AL(L+ 1). (5)

For a physical choice of parameters (h2, A > O), the lowest 5U(3) irrep is (2N, O) which hasrotational states with L = 0+,2+,4 +, ... , (2N)+ (I( = O) resembling the ground band ofan axial!y deformed nueleus. The first excited 5U(3) irrep (2N - 4, 2) contains both the (3(I( = O,L = 0+,2+,4+, ... ) and 'Y(I( = 2, L = 2+,3+,4+, ... ) bands. Consequently, statesin these bands with the same angular momentum are degenerate. This (3-'Ydegeneracy isa characteristic feature of the 5U(3) limit of the IBM which, however, is not commonlyobserved 12]. In most deformed nuelei the (3 band lies aboye the 'Yband as is evident fromthe experimental spectrum of 168Er shown in Fig. 1. In the IBI\! framework, with at mosttwo-body interactions, one is therefore compel!ed to break the 5U(3) symmetry in orderto conform with the experimental data. This can be done by ineluding in the Hamiltonianterms from other chains so as to lift the undesired (3-'Ydegeneracy. 5uch an approach wastaken in Ref. [3] where the Casimir of 0(6) was added to the 5U(3) Hamiltonian of Eq. (3).5imilarly, in the consistent Q formalism 141,the assumed quadrupole Hamiltonian involvesa particnlar mixture of Casimir operators of al! group chains in Eq. (1). Both treatmentsyield a satisfactory description of the spectroscopic data of 168Er below 2 MeV. However,in this procedure, the 5U(3) symmetry is completely broken, al! eigenstates are mixedand no analytic solutions are retained. The question we wish to explore is whether it ispossible to break the symrnetry but in a very particular way so that part of the states(but not al!!) wil! stil! be solvable with good symmetry. We wil! refer to this situation aspartial dynarnical symrnetry 15], to inclicate that the virtues of a dynamical syrnmetry (e.9.,solvability) are fulfil!ed but by only a subset of states. The construction of Harniltonianswith partial syrnmetries is non-trivial since such Hamiltonians cannot be scalars of thesymmetry group (otherwise al! states wil! have the symmetry). Nevertheless, an algorithmfor such a construction has been suggested [5]' and wil! be used in the discussion below.

PARTIALSYMMETRIESIN NUCLEARSPECTROSCOPY 155

E EXP SU(3) POS(MeV) -8'

-12- -12+

2.0 - -12+-i~

-6--e+-e+ -6-

-6--(6/

1.5 - -10. -10.

_'''10+-7+-4+-7- -7+-4+

-6+-2+ -6+-6+ -6+-2+-O- -O-

-5-1l -5- -5-1l1.0 - -4+ -4+-4+ -.--a+ -3+ -S-

-3--8'

-3--2- -2+-2+ -2-Y Y -O. Y

Il0.5 _ -6+ -S- -S-

-.- -.- -.--2- -2- -2-0.0 - -O. -O- -O-9 9 9

FIGURE l. Spectra of l6SEr. Experimental energies [3J(EXP) compared with an IBM calcnlation inan exact SU(3) dynamical symmetry (SU(3») and in a partial dynamical SU(3) symmetry (PDS).The latter employs the Hamiltonian in Eq. (13) wilh ho = 0.008, h, = 0.004, .\ = 0.013 MeV.

3. PARTIAL SU(3) SYMMETRY IN TIIE IBM

Consider the following rotational.invariant IBM Hamiltonian

H(ho,h2) = h2 [-CSU(3) + 2IV(2Ñ + 3)]+ (h2 - ho) [_4Ñ2 - 6Ñ + ñd - ñ~ + 4Ññd + 2CO(6) - CO(5)] , (6)

where ho, h2 are arbitrary constants and we use the definition of Casimir operators as inTable 1 of the Appendix in Ref. [61. Clearly, for ho f h2 the above Hamiltonian contains amixture of Casimir operators of all ehains in Eq. (1), henee it breaks the SU(3) symmetry.The specifie breaking, however, respects SU(3) as a partial symmetry. To prove this non-trivial statement, it is simpler to eonsider the normal arder-arder form

H(ho, "2) = hoPJPo + h2PJ . [>2= hoPJPo + h2 2::= pJ.~P2.",l'

(7)

156 A. LEVIATAN

where F2,1' = (_)1' P2,-w The Hamiltonian is seen to be constructed from boson palroperators with angular momentum L = O and 2, which are delined as

PJ = dI .dl - 2(81)2,

p,1 = 281dl + V7(dldl)(2) (8)2,Jl JJ IJ. •

These boson pair operators satisfy the following properties

PL,l'le; N) = O; L = 0,2,

[PL,I" PJ.2J le; N) = 6L,261',22(6N + 9)le;N),

[[PL,I" PJ.2J PJ.2J = 6L,261',2 24PJ.2'

The state le;N) in Eq. (9) is a condensate of N bosons and is given by

(9)

(10)

It is the lowest weight state in the SU(3) irrep (A, /1) = (2N, O) and serves as an intrinsicstate [7] for the SU(3) ground bando The rotationa! members of the ground band withgood angular momentum can be obtained by projection from le;N).A number of interesting observations follow from the relations in Eq. (9) and the struc-

ture of H(ho, h2). As is evident from Eq. (6), for h2 = ho, the Hamiltonian is related tothe SU(3) Casimir operator and thus becomes an SU(3) sealar. From the faet that theboson pair operators (8) transform as (0,2) under SU(3), it follows that for h2 = -ho/5the Hamiltonian transforms as a (2,2) SU(3) tensor component. For arbitrary ho, h2 eo-efficients, H(h2; ho) is therefore not an SU(3) sealar, nevertheless it always has le; N)(which has good SU(3) (2N, O) species) as an exact zero-energy eigenstate (for any N).This property is a direct outcome of the lirst relation in Eq. (9) and of the structure ofthe Hamiltonian (7). IBM Hamiltonians which have le; N) as an eigenstate were lirst en-countered in the study of the intrinsic structure of the model [6,81. They have the genericform of Eq. (7) and have provided the crucial guidelines for the introduction of partialsymmetries. Since H(ho, h2) is an 0(3) scalar, it follows that states of good L projectedfrom le;N) are also zero energy eigenstates which span the (2N, O) representation. Thus,when the coefficients ho, h2 are positive, the Hamiltonian (7) becomes positive delinite byconstruction, and even though it is not an SU(3) scalar, it still has an exactly degeneratezero-energy ground state band whose rotational members possess good SU(3) symmetry.Even more surprising is the presence of additional eigenstates in excited bands of H(ho, h2)with good SU(3) character. This is dile to the relations for single and dOllble commutatorsin Eq. (9), from which we lind that the sequence of states

Ik) ex (pi,2) k le; N - 2k), (11)

PARTIAL SYMMETRIES IN NUCLEAR SPECTROSCOPY 157

are eigenstates of H(ho, h2) in Eq. (7) with eigenvalues

Ek = 6h2 (2N + 1 - 2k) k. (12)

By comparillg the latter eigen-energies with the 5U(3) eigenvalues in Eq. (5), we concludethat these Ik) states are in the 5U(3) irreps (2N - 4k,2k) with 2k ~ N. lt can befurther shown that they are lowest weight states in these representations. The states Ik)are deformed and serve as intrinsic states representing -l bands with angular momentumprojection (I< = 2k) along the symmetry axis [9J. In particular, Ik = O) represents theground-state band (I< = O) and Ik = 1) is the ')'-band (I< = 2). The intrinsic states breakthe 0(3) symmetry but since the Hamiltonian in Eg. (7) is an 0(3) scalar, the projectedstates 1(2N - 4k, 2k) I< = 2k; L, M), with good L 2: I< are also eigenstates of H(ho, h2)with goud 5U(3) symmetry. lt shuuld be noted that fur k f Othe states projected from Ik)span only.part of the corresponding 5U(3) irreps. There are other states originally in theseirreps (as well as in other irreps) which do not preserve the 5U(3) symmetry and thereforeget mixed. This situation corresponds precisely to that of partial 5U(3) symmetry. AnHamiltonian H(ho, h2) which is not an 5U(3) scalar has a subset of solvable eigenstateswhich continue to have good 5U(3) symmetry. AII of the abuve discussion is applicablealso to the case when we add to the Hamiltonian (7) the Casimir operator of 0(3), and bydoing so converting the partial SU(3) symmetry into partial dynamical 5U(3) symmetry.The additional rotational term contributesjust an L(L+l) splitting but does not affect thewave functions. 5chematic numerical studies [51have shown that, in general, the breakingof the 5U(3) symmetry induced by the Hamiltonian (7) is not small and can lead to anappreciable mixing. Nevertheless, the special sulvable ,tates carry good 5U(3) labels. The5U(3) symmetry is thercfore partial but exact.

4. PARTlAL 5U(3) SYMMETRY IN DEFORMED NUCLEI

A relevant question at this stage is whether partial dynamical symmetries (pds) are justa formal notion or can they. actually be realized in nuclei. Ir so, they may serve as a usefultool in realistic applications to nuclear spectroscopy. In this sectiun we consider 168Er as atypical example uf an axially, deformed prolate nucleus in the rare earth regiun, and shuwthe relevance of 5U(3) pds to its description.

The experimental spectra [31 uf the ground (g), (3, and ')' ballds in 168Er is shown inFig. 1. \Ve attempt a descriptiun in terms uf an IBM lIamiltunian with partial dynamical5U(3) symmetry

(13)

According to the discussion in 5ect. 3, the spectrum of the ground and ')' bands is solvableand is given by

Eg(L) = ).,L(L + 1),

E,(L) = 6h2(2N - 1) + AL(L + 1). (14)

158 A. LEVIATAN

TABLE I. SU(3) decomposilion oC lowesl members oC lhe ground (g) '"Y and fJ bands in lhe parlía!dynamiea! SU(3) ealculalion oC Fig. 1.

(32,0) (28,2)0+ 1.092+ 1.09

2+ 1.0,3+ 1.0,0+ 0.87{J

2+ 0.87{J

(24,4)

0.030.03

(26,0)

0.1

0.1

The Hamillonian in Eq. (13) is spedfied by lhree paramelers (N = 16 for J6SEr accord-ing to the usual boson counting). \Ve extract the values of .\ and h2 fram the followingexperimental energy differences

.\= ~(E2+ - Eo+)'• •E2+ - E2+, .6(2N - 1) .

(15)

For an exact SU(3) dynamical symmetry, ho = h2 implying E{J(L) = E,(L) for L ~ 2,even. The corresponding spectrum in this case (shown in Fig. 1) deviates considerablyfram the experimental data sinee empirieally the fJ and 'Y bands are not degenerate.On the other hand, when the dynamical SU(3) symrnetry is partial, one can vary hoso as to reproduce the (3 bandhead energy E{J(L = O) (The large-N estimate [8] f{J =4N(2ho + h2) is a convenient initial guess). Having determined the three parameters .\,ho, h2, the prediction for other rotational members of the graund (3 and 'Ybands is shownin Fig. 1. Clearly, the SU(3) pds spectrum is an improvement over the schematic, exactSU(3) dynamical symmetry description, sinee the (3-'Y degeneracy is lifted. The goodSU(3) character for the graund and 'Ybands is retained in the pds calculation, while the(3 band is mixed. The SU(3) decomposition of selected states in these bands are shown inTable 1.

Electromagnetic transitions are a more sensitive probe to the structme of wave func-tions, hence are an important indicator for verifying the relevance of partial SU(3) sym-metry. To calculale such observables we need to spedfy lhe wave-functions of the initialand final states as well as the operator that induces lhe transition. For the Hamilto-nian in Eq. (13) with partial dynarnical SU(3) symllletry, the solvable states are thoseprajected fram the intrinsic slales Ih)k(2N - 41.:,2/,;)1{ = 21.:), and are simply selectedmembers of the ElIiott basis 1>d (.\, l' )I{ LM) [lO]. In particular, lhe stales belonging to lheground and 'Ybands are the ElIiott states 1>d(2N, 0)/\ = O, LM) and 1>d(2N - 4, 2)K =2, LM) respectively. Their wave functions can be exprcsscd in tcrms of thc Vergados basiswv((.\'¡')XLM) [11], which is thc usual (hut nol uniquc) choicc for orthonormal SU(3)

PARTIAL SYMMETRIES IN NUCLEAR SPECTROSCOPY 159

basis. Specifically,

(g) Leven: <pd(2N, O)K = O,LM) = Wv((2N,0)X = O,LM),

h) L odd: <pd(2N - 4, 2)K = 2, LM) = Wv((2N - 4, 2)X = 2, LM),

h) Leven: <P£((2N - 4, 2)K = 2LM) =

[ Wv((2N - 4, 2)X = 2, LM) - x~~)Wv((2N - 4, 2)X = O, LM) ] /xW. (16)

It should be noted that Elliott states in the -y(/< = 2) band with even values of L aremixtures of Vergados states in the {3(X = O) and -y(X = 2) bands. The x~~),x~~) arecoefficients which appear in the transformation between the two bases [111

(L) [ (L - l)L(L + l)(L + 2) ] 1/2x20 = - 4(2N - 2)(2N - 1)(2N - L - 2)(2N + L - 1)

x(L) = [!2(2N - 2)2 - L(L + 1)J[2(2N - 1)2 - L(L + 1)]] 1/2 (17)22 4(2N - 2)(2N - 1)(2N - L - 2)(2N + L - 1)

The most general 113Mone-body E2 operator may be written as

(18)

where Q(2) is the quadrupole SU(3) generator given in Eq. (4). The matrix elements ofsuch E2 operator in the Vergados basis are known in closed form [12,131. Using the explicitexpressions of the wave functions in Eq. (16), it is thercfore possible to obtain analyticexpressions for the E2 rates between the subset of solvable states. Thus, for intrabandtransitions in the (2N,0)I< = Oground band (L even, L' = L:I: 2), we get

B£(E2; gI< = O, L - gI< = O, L') = Bv(E2;!lX = O, L - 9X = O, L'). (19)

For interband trallsitions between the (2N - 4, 2)I< = 2, -y band and the (2N,0)I< = O,ground band (L odd, L' = L:I: 1) we get

Be(E2; -yI< = 2, L - gI< = O, L') = Bv(E2; 'YX = 2, L - 9X = O, L'), (20)

and for Leven, L' = L, L :l: 2 we have

B£(E2; -yI< = 2, L - gI< = O, L') =

[ JBv(E2;'YX = 2,L -!IX = 0,£1) :l: x~~)J Bv(E2; {3x = O, L - !IX = O, U) ] 2 ( )(L) , 21

x22

where the + (-) sigll applies to a transition with L' = L (L' = L:l: 2). In Eqs. (19)-(21)the notation Bv(E2) and Be(E2) stands for B(E2) vallles eaklllated in the Vergados and

160 A. LEVIATAN

TABLE IJ. B(E2) values [13] for 9 - 9, -y - 9 and {3- 9 transitions, calculated in the Vergadosbasis with the classification: [(2N,0)X = O,LJ for the ground (9) band [(2N -4,2)X = 2,L] for the-y band [(2N - 4, 2)X = 0, L] for the {3bando The E2 operator is given in Eq. (18).

Transition

9L - 9L - 2

-yL-9L-2

"(L - 9L - 1

{3L - 9L - 2

{3L - 9L

Bv(E2)

2N2 [ + (4N-Ij 0]2 3(L-I)L (2N-L+2)(2N+L+l)Q 3(2N-1) 2(2L-l)(2L+1) 4NJ

'N02 ~ !2N-L-I)(2N-L+l)(2N+L)3 2(2L+ 1) 2N(2N -3)(2N -1)

'dN82 L-l (2N-L-l)(2N+L)(2N+L+2)3 2(2L+ 1) 2N(2N -3)(2N - ¡)

2NfP {L-l)L 2( N -1 )(2¡\.' -L-2){2N+L-l )(2N-f L+ 1)(2,\1 +L+3)3 4(2L+l)(2L+3) N(21V-3)(2N-l)[8(N-l)2-L(L+Q]

2I N(P 3(L-I)L [4(N-1)2+L+1] [ (2N-L)(2N-L+2) ]'9 2(2L-l)(2L+l) 2N-1 N(2N-3)[8(N-l)2-L(L+I)]

!N(J2 L(L+I) [4(N-1)1_2L(L+1)+3]2 [ (2N-L~2N+L+l) ]9 (2L-l)(2L+3) 2N-I N(2N-3)18( -l)2-L(L+l)J

El!iott bases respectively. Analytic expressions for IJv(E2) values have been derived [131,and for completeness, are reproduced in Table IJ.In general the parameters a and e of the E2 operator in Eq. (18) can be extracted from

the experimental values of IJ(E2;Ot ~ 2t) and IJ(E2;Ot ~ 2~). The correspondingratio for 168Er is e/a = 4.256. As shown in Table IIJ, the resulting SU(3) pds E2 ratesfor 168Er are found to be in agreement with experiment and are similar to the calculationby Casten Warner amI Davidson [3] (where the SU(3) symmetry is broken for al! states).The SU(3) pds calculation reproduces correctly the ("( ~ "()/("( ~ g) strengths and thedominance of {3~ "( over {3~ 9 transitions. If we reeal! that only the ground band hasSU(3) component. ('\, ¡,) = (2N, O) (see Table 1) and that Q(2) in Eq. (18) is a generator ofSU(3) (henee eannot eonneet different ('\, ¡,) irreps), it fol!ows that {3,"( ~ 9 IJ(E2) ratiosare independent of both a and 8. Furthermore, sinee the ground amI "( bands have pureSU(3) charaeter, (2N, O) and (2N - 4,2) respectively, the eorresponding wave-funetions donot depend on parameters of the Hamiltonian henee are detennined solely by symmetry.

PARTIALSYMMETRIESIN NUCLEARSPECTROSCOPY 161

TABLE III. 8(E2) branehing ratios fram states in the 'Y band in l6SEr. The experimental ratios(EXP) and the braken SU(3) ealeulation of Warner Casten and Davidson (WCD) are taken framReí. [3J.(PDS) are the partial dynamieal SU(3) symmetry ealeulation reported in the present work.

r r EXP PDS WCD r r EXP PDS WCD• [ • [

2+ 0+ 54.0 64.27 66.0 6+ 4+ 0.44 0.89 0.97, 9 , 9

2+ 100.0 100.0 100.0 6+ 3.8 4.38 4.39 9

4+ 6.8 6.26 6.0 8+ 1.4 0.79 0.739 9

3+ 2+ 2.6 2.70 2.7 4~. 100.0 100.0 100.0, 9

4+ 1.7 1.33 1.3 5+ 69.0 58.61 59.09 ,2+ 100.0 100.0 100.0 7+ 6+ 0.74 2.62 2.7, , 9

4+ 2+ 1.6 2.39 2.5 5+ 100.0 100.0 100.0, 9 ,4+ 8.1 8.52 8.3 6+ 59.0 39.22 39.09 ,6+ 1.1 1.07 1.0 8+ 6+ 1.8 0.59 0.679 , 9

2+ 100.0 100.0 100.0 8+ 5.1 3.57 3.5, 9

5+ 4+ 2.91 4.15 4.3 6+ 100.0 100.0 100.0, 9 ,6+ 3.6 3.31 3.1 7+ 135.0 28.64 29.09 ,3+ 100.0 100.0 100.0,4+ 122.0 98.22 98.5,

Consequently, the B(E2) ratios for 'Y ~ 9 transitions quoted in Table III are parameter-free predietions of SU(3) pds. The agreement. bet.ween t.hese prediet.ions and t.he dat.aeonfirms t.he relevanee of part.ial dynamical SU(3) symmet.ry t.o t.he spect.roscopy of 16BEr.

5. SUMMARY

The not.ion of part.ial dynamical symmetry (pds) generalizes t.he familiar concepts of exact.and dynamical symmet.ries. In going fram an exact t.o a dynamical symmet.ry, the formerdegenerat.e st.at.es are split. but. not. mixed, and t.he block structure of t.he Hamiltonian isret.ained. Proceeding t.o partial symmet.ry, some blocks (or select.ed states in a given block)remain pure, while ot.her st.at.es mix and loose t.he symmet.ry charact.er.\Ve have const.ruct.ed explicit.ly mM Hamiltonians with SU(3) pds. Such Hamiltonians

are not. invariant. under SU(3) but. have a subset. of eigenstat.es wit.h good SU(3) symmet.ry.The special st.at.es are solvable and span part. of part.icular SU(3) irreps. Their wave-funct.ions, eigenvalues and E2 rat.es are known analyt.ically. An applicat.ion of t.he schemet.o 16BEr has demonst.rat.ed that. the empirical spectrum and E2 rat.es conform with thepredietions of partial SU(3) symmetry. These observations point at the relevanee of partialSU(3) symmet.ry to the spectroscopy ofaxially deformed nuelei, at least, as a start.iug pointfor further refinements.

162 A. LEVIATAN

The notion of partial dynamical symmetry (pds) is not confined to SU(3). A general al-gorithm is available for constructing Hamiltonians with pds for any semi-simple group [51.The occurrence of partial (but exact) symmetries imply that part of the eigenvalues andwave functions can be found analytical!y but not the entire spectrum. As such, pds canovercome the schematic features of exact dynamical symmetries (e.g., undesired degen-eracies) and simultaneously retain their virtues (i.e., solvability) for sorne states. Pds canaddress situations where a subset of levels exhibit a symmetry which is not shared byal! states and therefore does not arise from invariance of the Hamiltonian. It remains tobe seen whether these attributes can transform the mathematical notion of pds into aworking tool for practical applications in realistic systems.

Hamiltonians with partial symmetries are not completely integrable and may exhibitchaotic behavior. This makes them a uscful tool to study mixed systems with coexistingregularity and chaos. The interest in mixed systems is driven by their being generic [141,i.e., most realistic systems are neither ful!y chaotic nor ful!y regular. An initial inves-tigation revealed that partial symmetries may cause suppression of chaos even in caseswhere the fraction of solvable states vanishes in the classical limit [15]. It wil! be of greatinterest to explore the ramifications of partial symmetries both for discrete spectroscopyand statistical aspects of nuclei.

ACKNOWLEDGMENTS

This research was supported by the Israel Sciell{:e Foundation administered by the IsraelAcademy of Sciences and Humanities.

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